The result based on unofficial tallies, if confirmed for this round and repeated in Saturday's second stage, may give Islamist President Mohamed Morsi limited cause for celebration as it shows the wide rift in Egypt at a time when he needs to build consensus on tough measures to heal a fragile economy.
Official results are not expected till after the next round.
Morsi and his backers say the constitution is vital to move Egypt's democratic transition forward. Opponents say the basic law is too Islamist and tramples on minority rights, including those of Christians who make up 10 percent of the population.

The build-up to Saturday's vote was marred by deadly protests. Demonstrations erupted last month when Morsi awarded himself sweeping powers and then fast-tracked the constitution through an assembly dominated by his Islamist allies.
The vote passed off peacefully with long queues forming in Cairo and other cities and towns where this round of voting was held. The vote was staggered because many judges needed to oversee polling staged a boycott to voice their opposition.
But late on Saturday, as polls were closing, Islamists attacked the offices of the liberal opposition Wafd party newspaper, a party that was part of the National Salvation Front coalition that pushed for a "no" vote.
"The referendum was 56.5 percent for the 'yes' vote," a senior official in the Brotherhood's Freedom and Justice Party operations room set up to monitor voting told Reuters.

The Brotherhood and its party, which propelled Morsi to power in a June election, had representatives at almost all polling stations across the 10 areas, including Cairo, where this round of voting was held.
The official, who asked not to be identified, said the tally was based on counts from more than 99 percent of polling stations in this round.
'Very close'One opposition official also said the vote appeared to have gone in favor of Islamists who backed the constitution, after the opposition had previously said late on Saturday when voting ended that their exit polls indicated the "no" camp would win.
Another opposition official had suggested as counting proceeded through the night that the vote would be "very close."
Even a narrow loss could hearten leftists, socialists, Christians and more liberal-minded Muslims who make up the disparate opposition camp, which has been beaten in two elections since Hosni Mubarak was overthrown last year.
They were drawn together to oppose what they saw as Morsi's power grab and his constitution push. Their National Salvation Front includes prominent figures such as Nobel Peace Prize laureate Mohamed ElBaradei, former Arab League chief Amr Moussa and firebrand leftist Hamdeen Sabahy.

If the constitution is approved, a parliamentary election will follow early next year. Opposition leaders say the Front could help unite the opposition for that poll after their divided ranks have split the vote in previous elections.
But analysts questions whether the group in this form will survive to a parliamentary election. The Islamist-dominated lower house of parliament elected earlier this year was dissolved based on a court order in June.
Violence in Cairo and other cities has plagued the run-up to the referendum. At least eight people were killed when rival camps clashed during demonstrations outside the presidential palace earlier this month.
Several party buildings belonging to the Muslim Brotherhood's party have been burned by angry protesters.
'Move on'On Friday, a day before the vote, rival factions armed with clubs, knives and swords fought in the streets of Alexandria. Opposition supporters trapped a Muslim preacher inside his mosque after he called for a "yes" vote.
"The sheikhs (preachers) told us to say 'yes' and I have read the constitution and I liked it," said Adel Imam, 53, as he queued to vote in Cairo on Saturday. "The country will move on."
Echoing the views of many Christians, Michael Nour, a 45-year-old Christian teacher in Alexandria, said: "I voted 'no' to the constitution out of patriotic duty. The constitution does not represent all Egyptians."
In order to pass, the constitution must be approved by more than 50 percent of voters who cast ballots. A little more than half of Egypt's electorate of 51 million were eligible to vote in the first round.

Rights groups reported some abuses, such as polling stations opening late, officials telling people to vote "yes," bribery and intimidation.
But Gamal Eid, head of the Arab Network for Human Rights Information, which is monitoring the vote, said nothing reported so far was serious enough to invalidate the referendum.
Islamists have been counting on their disciplined ranks of supporters and the many Egyptians desperate for an end to turmoil that has hammered the economy.
Howaida Abdel Azeem, a post office employee, said: "I said 'yes' because I want the destruction the country is living through to be over and the crisis to pass."
If the constitution is voted down, a new assembly will have to be formed to draft a revised version, a process that could take up to nine months.
The army has deployed about 120,000 troops and 6,000 tanks and armored vehicles to protect polling stations and other government buildings. While the military backed Mubarak and his predecessors, it has not intervened in the present crisis.